Many dynamical systems in engineering and natural sciences exhibit “chaotic” behavior: their trajectories appear disordered and they are highly sensitive to initial data. The simplest mathematical examples of such systems are uniformly hyperbolic and uniformly expanding, and so hyperbolic dynamical systems have been at the forefront of smooth ergodic theory since at least the 1960s. However, most stochastic dynamical systems arising from physical and natural phenomena are not uniformly hyperbolic. In these instances, uniqueness results for uniformly hyperbolic dynamical systems and surface maps (such as those described in [1]}) no longer apply. Examples of such dynamical systems include the Lorenz attractor model of atmospheric convection, the associated geometric Lorenz attractor (described in more detail below), and the Belykh attractor of phase synchronization theory [2]}, [3]}, [4]}. The latter two are maps of the unit square that admit highly complex limit sets, so that the resulting maps are not invariant under Lebesgue volume and may not a priori admit a unique SRB measure.
